### Abstract

In this paper, we investigate the commutative algebra of the cohomology ring H* (G, k) of a finite group G over a field k. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo-Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler.

As a by-product of this investigation, we give a new proof of the Bourguiba-Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra.

Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.

Original language | English |
---|---|

Pages (from-to) | 171-197 |

Number of pages | 26 |

Journal | Illinois Journal of Mathematics |

Volume | 48 |

Issue number | 1 |

Publication status | Published - Mar 2004 |

### Keywords

- RESOLUTIONS
- ALGEBRA
- DEPTH

### Cite this

*Illinois Journal of Mathematics*,

*48*(1), 171-197.

**Dickson invariants, regularity and computation in group cohomology.** / Benson, David John.

Research output: Contribution to journal › Article

*Illinois Journal of Mathematics*, vol. 48, no. 1, pp. 171-197.

}

TY - JOUR

T1 - Dickson invariants, regularity and computation in group cohomology

AU - Benson, David John

PY - 2004/3

Y1 - 2004/3

N2 - In this paper, we investigate the commutative algebra of the cohomology ring H* (G, k) of a finite group G over a field k. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo-Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler.As a by-product of this investigation, we give a new proof of the Bourguiba-Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra.Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.

AB - In this paper, we investigate the commutative algebra of the cohomology ring H* (G, k) of a finite group G over a field k. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo-Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler.As a by-product of this investigation, we give a new proof of the Bourguiba-Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra.Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.

KW - RESOLUTIONS

KW - ALGEBRA

KW - DEPTH

M3 - Article

VL - 48

SP - 171

EP - 197

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

SN - 0019-2082

IS - 1

ER -